Study on the Infrared and Raman spectra of Ti3AlB2, Zr3AlB2, Hf3AlB2, and Ta3AlB2 by first-principles calculations

In this paper, the crystal geometry, electronic structure, lattice vibration, Infrared and Raman spectra of ternary layered borides M3AlB2 (M = Ti, Zr, Hf, Ta) are studied by using first principles calculation method based on the density functional theory. The electronic structure of M3AlB2 indicates that they are all electrical conductors, and the d orbitals of Ti, Zr, Hf, and Ta occupy most of the bottom of the conduction band and most of the top of the valence band. Al and B have lower contributions near their Fermi level. The lightweight and stronger chemical bonds of atom B are important factors that correspond to higher levels of peak positions in the Infrared and Raman spectra. However, the vibration frequencies, phonon density of states, and peak positions of Infrared and Raman spectra are significantly lower because of heavier masses and weaker chemical bonds for M and Al atoms. And, there are 6 Infrared active modes A2u and E1u, and 7 Raman active modes, namely A1g, E2g, and E1g corresponding to different vibration frequencies in M3AlB2. Furthermore, the Infrared and Raman spectra of M3AlB2 were obtained respectively, which intuitively provided a reliable Infrared and Raman vibration position and intensity theoretical basis for the experimental study.


Method and computational details
All first-principles calculations were performed by Vienna Ab-initio Simulation Package (VASP) with Plane-wave DFT methodology in this study 47 .The use of different exchange correlation functionals has been demonstrated to have a minor influence on the formation enthalpy [48][49][50][51] .And, the electronic exchange-correlation interactions were described by the generalized gradient approximation (GGA) with Perdew Burke Ernzerhof (PBE) functional [52][53][54][55] in the paper.
In the calculation process, a plane wave cut-off energy of 700 eV was utilized.Additionally, integrations in the Brillouin zone were conducted using a 15 × 15 × 2 Monkhorst-Pack as specific k-points, while a K-mesh of 0.03 was chosen throughout the calculation 56 .Energy and force convergences are 1 × 10 -8 eV and − 1 × 10 -7 eV/Å.In calculations, the Density Functional Perturbation Theory (DFPT) approach was employed in both Phonopy 57,58 and VASP to determine the second-order Interatomic Force Constants (IFCs).DFPT is a method for calculating the response of physical quantities to external fields using linear response theory.It can be used to calculate the force constant matrix.The Phono3py 59,60 code generates the displacement structure with irreducible representation along the center q-point to produce positive and negative displacement structures.Finally, the dielectric constant is calculated.Only the structure with irreducible representation is selected for dielectric constant calculation to reduce the calculation burden 61,62 .And Phonopy-Spectroscopy software were utilized in the procedures to calculate the IR and Raman spectra 63 .Hessian matrix data and born effective charges also have to be taken into account.Moreover, the focus of this research was primarily on the atom vibration image of the structure's Γ point active Infrared and Raman spectra associated to phonons.Additionally, 3d 2 4s 2 for Ti, 4d 2 5s 2 for Zr, 5d 2 6s 2 for Hf, 5d 3 6s 2 for Ta, 3s 2 3p 1 for Al, and 2s 2 2p 1 for B were considered as valence states in the paper.The software packages Pymatgen and Vaspvis were used in the study 64,65 .
Lattice dynamics has played a significant role in understanding certain physical characteristics of Infrared and Raman spectra.We employ the Phonopy package and VASP to obtain phonon dispersion and phonon density of states corresponding to lattice vibration in the paper.The phonon vibrational frequencies via DFPT method.Phonon frequencies and eigenvectors are computed in the harmonic limit using the second-order force constant matrix Φ αβ (jl, j′l′), the elements of which are the change in force in the α Cartesian direction acting on atom j in unit cell l, in response to the displacement of atom j′ in unit cell l′ in the β direction.The phonon frequencies are the eigenvalues of the dynamical matrix at a given wavevector q: m j refer to the atomic masses.The long-rang Coulomb interactions lead to non-analytic corrections in the limit q → 0, which cause the splitting of the longitudinal optic (LO) and transverse optic (TO) modes 66 : is the derivative of the cell polarization along the Cartesian direction (x, y, z) α with respect to the atomic displacement of atom i along β and defined as The dynamical charges Z* are called "Born effective-charge tensors", and can be calculated using perturbation theory.The calculations have been performed by using the LESPILON = True keyword of VASP.r is the displacement of the atom i. Ω is the volume of the cell and e is the electronic charge 67 .The Z * tensors were computed using DFPT as implemented in VASP.Ω represents the volume of the primitive unit cell, and e stand for the elementary charge constant.Within the dipole approximation, the Infrared intensity of an eigenmode s can be expressed as the square of the Born effective-charge and eignevector X β s, j as: (1) www.nature.com/scientificreports/X β s, j represents the normalized vibrational eigenvector of the jth phonon mode of the sth atom in the unit cell.Raman activity tensors of an eigenmode are evaluated by taking the derivative of the high-frequency macroscopic dielectric constant ε ∞ with respect to the normal mode amplitude Q(s) using a central difference scheme is: ε ∞ can be calculated by DFPT method and can be found in the OUTCAR with careful identification.At the same time, LESPILON = True is necessary.Q is the normal-mode coordinate at the Γ-point and is defined by u(s, j) represents the atomic displacement.Phonopy and Phonopy-Spectroscopy were used to compute phonon frequencies, as well as Raman intensities 68,69 .The Raman intensities are then defined as: In Eq. ( 8), the notation I αβ was used in place of I Raman,αβ (s) .With m j in amu, and Z* in e, the calculated IR intensities will have units e 2 amu −1 .With Q in amu 1/2 Å, the elements of the Raman-activity tensor have units of Å 2 amu −1/2 , and the units of the scalar Raman intensity.And the square of the Raman activity are Å 4 amu −1 at last 63,70 .

Structure
In Fig. 1, it is shown that the schematic graphs of crystal structure for M 3 AlB 2 , all of which belong to the P6 3 /mmc (No. 194) space group.Each unit cell contains two primitive cells, resulting in a total of 12 atoms in each conventional unit cell.And the compounds Ti 3 AlB 2 , Zr 3 AlB 2 , Hf 3 AlB 2 , and Ta 3 AlB 2 can be considered as 312 phases in the M n+1 AlB n phase structure, which is consistent with references 71,72 .In the M 3 AlB 2 structure, the Al layer atoms are inserted between the adjacent M layer atoms, and the B atoms fill the octahedral interstitial positions of the former.The Al atom is located at the center of the quadrangular prism, surrounded by the M-layer atoms.
Because the space at this position is larger than the octahedral gap, it can accommodate larger atoms such as Al.

Lattice constant
As shown in Table 1, the lattice constants of Ti 3 AlB 2 , Zr 3 AlB 2 , Hf 3 AlB 2 , and Ta 3 AlB 2 have been calculated.It can be observed that the calculation results of Ti 3 AlB 2 are in good agreement with the reference results 72 .
In general, the lattice constants "a" and "c" of the M 3 AlB 2 phase decrease with the increase in the atomic radius of the transition metal element M, but have little relationship with the number of d electron layers.The lattice (4) www.nature.com/scientificreports/constant of the 3d compound is significantly smaller than that of the 4d and 5d compounds, but the difference between the 4d and 5d compounds is minimal.At the same time, for a series of compounds with a given M 3 AlB 2 value, the c/a ratio falls within a very narrow range and is less affected by M atoms.We assume that the MAX phase is an interstitial compound, in which A and X atoms fill the interstices between M atoms.In the 312 phase, the c-axis direction comprises 6 layers of M atoms.Without considering the ideal lattice distortion caused by the addition of A and X atoms, the c-axis length should be 6 times that of the a-axis length.That is to say, the c/a ratio of the 312 phase should be 6, and other researchers have also found a similar trend in their investigation 34 .However, in reality, the presence of Al and B atoms inevitably impacts the lattice, causing the actual c/a ratio to deviate slightly from the ideal value.As shown in Table 1, the c/a ratios of Ti 3 AlB 2 , Zr 3 AlB 2 , Hf 3 AlB 2 , and Ta 3 AlB 2 are 6.231742, 6.114347, 6.074749, and 6.143366, respectively.This aligns with our speculation closely.

Electronic band structure
The projected band structures and density of states near the Fermi energy (E f ) for Ti 3 AlB 2 , Zr 3 AlB 2 , Hf 3 AlB 2 , and Ta 3 AlB 2 is depicted in Fig. 2. It is reveal that the energy bands of these compounds intersect the Fermi surface, suggesting that Ti 3 AlB 2 , Zr 3 AlB 2 , Hf 3 AlB 2 , and Ta 3 AlB 2 exhibit electrical conductivity 9 .This has typical characteristics of MAX phase.It can be seen from the energy band that Ti, Zr, Hf, and Ta are dominant in Ti 3 AlB 2 , Zr 3 AlB 2 , Hf 3 AlB 2 , and Ta 3 AlB 2 materials near the Fermi level.It is also evident that the d orbitals of Ti, Zr, and Hf predominantly occupy the lower portion of the conduction band and the upper portion of the valence band.Al has a minor contribution at the top of the valence band, while it has a very small impact at the bottom of the conduction band, which is almost negligible.B also makes a specific contribution at the bottom of the conduction band and the top of the valence band, primarily in the p orbital.The Fermi level in Ta 3 AlB 2 is predominantly occupied by the d orbital of the Ta element, while the contribution of Al and B near the Fermi level of Ta 3 AlB 2 is minimal and negligible.
It can be observed from these figures that the fundamental characteristics in the density of states for the same series of compounds, Ti 3 AlB 2 , Zr 3 AlB 2 , Hf 3 AlB 2 , and Ta 3 AlB 2 , are very similar.Since it is far from the Al atom, the d orbitals of M atom primarily hybridize with the B-p state, and there is some hybrid state with Al atoms.Thus, the M atom forms a covalent bond with the B atom.At the same time, these compounds contain weak M-M covalent bonds.The hybrid state is in a high-energy range, the M-M bond is weaker than the M-B bond.
Combined with Fig. 3, it can be observed that the lower energy band primarily originates from the B-s state.The spikes in this range are primarily caused by the strong hybridization between the M-d state and the B-p state, specifically the formation of the M-B covalent bond.The energy band at the top of the valence band is the result of hybridization between the M-d state and the Al-p state, which corresponds to M-Al covalent bond.The energy range corresponding to the M-B bond is significantly lower than that of the M-M bond.And, M-B, M-Al bond is stronger than M-M bond.
As shown in Fig. 3

Lattice vibration
The lattice vibration spectra of Ti 3 AlB 2 , Zr 3 AlB 2 , Hf 3 AlB 2 , and Ta 3 AlB 2 are depicted along the direction of high symmetry points in the Brillouin zone as shown in Fig. 4. Firstly, there is no phonon vibration with negative frequency in the phonon dispersion in the studied compounds, which indicates that there is no imaginary frequency and Ti 3 AlB 2 , Zr 3 AlB 2 , Hf 3 AlB 2 , and Ta 3 AlB 2 are stable.Secondly, it can be observed from these figures that Ti 3 AlB 2 , Zr 3 AlB 2 , Hf 3 AlB 2 , and Ta 3 AlB 2 compounds have phonon band gaps.For Ti 3 AlB 2 , the phonon band gap is 306.97 cm −1 -465.33 cm −1 .To Zr 3 AlB 2 , the phonon band gap is 276.67 cm −1 -406.63 cm −1 .And there are phonon band gaps of 277.33 cm −1 -441.66 cm −1 and 340.66 cm −1 -589.00 cm −1 for Hf 3 AlB 2 , and Ta 3 AlB 2 respectively.Furthermore, the phonon vibrations of M and Al are below the phonon band gap, while the phonon vibrations above belong to B atoms.Similar to the electronic state, the flat band in the phonon spectrum corresponds to the peak in the phonon state density as shown in  Thirdly, compared to M and Al atoms, the mass of B atoms is significantly lower.More importantly, B atoms are connected to the lattice through very strong M-B bonds.This results in the vibration frequency of the B atom being significantly higher than that of other atoms.The phonon energy intervals corresponding to M and Al atoms are more complex.On the one hand, based on the electronic structure and chemical bonding mentioned above, M atoms are connected to the lattice through strong M-B bonds and weak M-Al bonds.Therefore, the binding of M atoms to the lattice should be stronger than that of Al atoms.On the other hand, the M atom is much heavier than the Al atom.These factors result in the phonon states corresponding to M and Al atoms being at lower energy levels, but the relative energy level changes of the two are more complicated.And the B atoms are at higher energy levels and form phonon band gaps with them.
In the M 3 AlB 2 compound, the phonon density of states (PHDOS) below the phonon band gap primarily corresponds to the lattice vibration dynamics of M and Al atoms, while above the band gap, it relates to the lattice vibration of B atoms.There is a sharp peak in the middle in the PHDOS of Zr  The acoustic branch has two irreducible representation.E 1u is doubly degenerate, and the acoustic branch contains 3 lattice waves.And the acoustic branch is as following Eq.( 10): There are 22 types of irreducible representation in the optical branch, among which E 1g , E 2g , E 1u , and E 2u are doubly degenerate, corresponding to 33 lattice waves.The optical branch is as following Eq.( 11): Therefore, there are a total of 36 lattice waves.And Table 2 presents the frequencies of phonon vibration modes at the center of the Brillouin zones for Ti 3 AlB 2 , Zr 3 AlB 2 , Hf 3 AlB 2 , and Ta 3 AlB 2 .E 1g , E 2g , E 1u , and E 2u are doubly degenerate, and the frequencies of the corresponding phonon vibration modes are identical.
As shown in Table 2, the highest phonon frequency of Ti 3 AlB 2 is approximately 609.0 cm −1 , while Zr 3 AlB 2 has a highest phonon frequency of about 536.5 cm −1 .Hf 3 AlB 2 exhibits a highest phonon frequency of around 591.1 cm −1 , and Ta 3 AlB 2 has the highest phonon frequency at approximately 673.2 cm −1 .Ti 3 AlB 2 , Zr 3 AlB 2 , Hf 3 AlB 2 , and Ta 3 AlB 2 .Although the atom mass of Ti, Zr, Hf, and Ta increases in turn, the highest phonon frequency may be related to atom mass, bond length, and bond angle.Therefore, the highest phonon frequency of Ti 3 AlB 2 , Zr 3 AlB 2 , Hf 3 AlB 2 , and Ta 3 AlB 2 does not increase completely in accordance with the atom mass of Ti, Zr, Hf, and Ta.It can be seen from the table that for similar crystal structures (Ti 3 AlB 2 , Zr 3 AlB 2 , Hf 3 AlB 2 , and Ta 3 AlB 2 ), the Mulliken symbol order corresponding to the characteristic table is not necessarily the same.This discrepancy may be attributed to the distinct nature of the Ti, Zr, Hf, and Ta elements, leading to differences in the fine structure of the crystal.This may be one of the reasons why people use Raman spectroscopy as a "fingerprint spectrum" to distinguish between different crystals and even very similar crystal fine structures.
According to the character       9a, it can also be seen that Ti 3 AlB 2 has 7 Raman active modes: E 2g (83.9)R,E 1g (107.8)R,A 1g (142.9)R,E 2g (192.2)R,E 1g (662.7)R,E 2g (663.7)R,A 1g (673.2)R.Figure 9c gives their specific peak position and intensity.From Fig. 9c, it can be seen that all of peaks of Raman spectra in Ta 3 AlB 2 have high vibration intensity, while E 1g (662.7)R,E 2g (663.7)R is very close and need to be carefully distinguished in the experiment.

Conclusion
In summary, the study provided detailed insights into the structural, electronic, and mechanical properties of M 3 AlB 2 compounds using first-principles calculations.The results contribute to the understanding of these materials and their potential applications in various fields.The findings can guide further experimental investigations

2 )
, we find that all the d orbitals (d xy , d xz , d yz , d x 2 -y 2 , and d z 2 ) in Ti near the Fermi surface of Ti 3 AlB 2 are given from the band and projected density of states of Ti 3 AlB 2 , Zr 3 AlB 2 , Hf 3 AlB 2 , and Ta 3 AlB 2 .And the contribution of Ti(d yz ) and Ti(d x 2 -y 2 ) is particularly pronounced.The contribution of Al is very small here, and only p z of B contributes to the density of states.All the d orbitals (d xy , d xz , d yz , d x 2 -y 2 , and d z 2 ) in Zr near the Fermi surface of Zr 3 AlB 2 contribute.Among these, Zr(d yz ) and Ti(d x 2 -y 2 ) contribute more significantly, while Al contributes very little.Additionally, only the p z orbital of B contributes here.All the d orbitals (d xy , d xz , d yz , d in Hf near the Fermi surface of Hf 3 AlB 2 contribute.Among these, Hf(d yz ) and Ti(d significantly, while Al contributes very little.B has p y and p z orbitals that contribute, with a larger contribution.All the d orbitals (d xy , d xz , d yz , d x 2 -y 2 , and d z 2 ) in Ta near the Fermi surface of Ta 3 AlB 2 contribute, with Ta(d and Ta(d xz ) making the most significant contributions, while Al and B contribute minimally.

Fig. 5 .
Fig. 5.This indicates that the lattice vibration exhibits clear localization characteristics.The frequencies of the optical branches near the long-wave (Γ point) are significantly different, indicating a noticeable ionic character in these compounds.Thirdly, compared to M and Al atoms, the mass of B atoms is significantly lower.More importantly, B atoms are connected to the lattice through very strong M-B bonds.This results in the vibration frequency of the B atom being significantly higher than that of other atoms.The phonon energy intervals corresponding to M and Al atoms are more complex.On the one hand, based on the electronic structure and chemical bonding mentioned above, M atoms are connected to the lattice through strong M-B bonds and weak M-Al bonds.Therefore, the binding of M atoms to the lattice should be stronger than that of Al atoms.On the other hand, the M atom is much heavier than the Al atom.These factors result in the phonon states corresponding to M and Al atoms being at lower energy levels, but the relative energy level changes of the two are more complicated.And the B atoms are at higher energy levels and form phonon band gaps with them.In the M 3 AlB 2 compound, the phonon density of states (PHDOS) below the phonon band gap primarily corresponds to the lattice vibration dynamics of M and Al atoms, while above the band gap, it relates to the lattice vibration of B atoms.There is a sharp peak in the middle in the PHDOS of Zr 3 AlB 2 , Hf 3 AlB 2 , and Ta 3 AlB 2 due to the vibration of Al atoms.And the proportion of Ti, Zr, Hf, and Ta in PHDOS of Ti 3 AlB 2 , Zr 3 AlB 2 , Hf 3 AlB 2 , and Ta 3 AlB 2 respectively, increases at low frequencies.It is assumed that an increase in the number of d electron layers results in and its atom mass, leading to a decrease in the lattice vibration frequency for M atom.For Al and B atoms, although the changing trend of the phonon frequency of the Al atom is similar to that of the M atom, the decisive factor should be the bond strength of M-Al, and the changing trend of the phonon frequency of the B atom is determined by the bond strength of M-B.Among the M 3 AlB 2 phase compounds, M-Al bond exhibits the lowest chemical bond stiffness in each system, with a stiffness value approximately 1/3-1/2 of the corresponding M-B bond stiffness.Additionally, the M-B bond closest to the Al atom demonstrates the highest stiffness.The results in Fig. 5 also indirectly indicate the changing trend of M-Al bond and M-B bond strength.

3
Fig. 5.This indicates that the lattice vibration exhibits clear localization characteristics.The frequencies of the optical branches near the long-wave (Γ point) are significantly different, indicating a noticeable ionic character in these compounds.Thirdly, compared to M and Al atoms, the mass of B atoms is significantly lower.More importantly, B atoms are connected to the lattice through very strong M-B bonds.This results in the vibration frequency of the B atom being significantly higher than that of other atoms.The phonon energy intervals corresponding to M and Al atoms are more complex.On the one hand, based on the electronic structure and chemical bonding mentioned above, M atoms are connected to the lattice through strong M-B bonds and weak M-Al bonds.Therefore, the binding of M atoms to the lattice should be stronger than that of Al atoms.On the other hand, the M atom is much heavier than the Al atom.These factors result in the phonon states corresponding to M and Al atoms being at lower energy levels, but the relative energy level changes of the two are more complicated.And the B atoms are at higher energy levels and form phonon band gaps with them.In the M 3 AlB 2 compound, the phonon density of states (PHDOS) below the phonon band gap primarily corresponds to the lattice vibration dynamics of M and Al atoms, while above the band gap, it relates to the lattice vibration of B atoms.There is a sharp peak in the middle in the PHDOS of Zr 3 AlB 2 , Hf 3 AlB 2 , and Ta 3 AlB 2 due to the vibration of Al atoms.And the proportion of Ti, Zr, Hf, and Ta in PHDOS of Ti 3 AlB 2 , Zr 3 AlB 2 , Hf 3 AlB 2 , and Ta 3 AlB 2 respectively, increases at low frequencies.It is assumed that an increase in the number of d electron layers results in and its atom mass, leading to a decrease in the lattice vibration frequency for M atom.For Al and B atoms, although the changing trend of the phonon frequency of the Al atom is similar to that of the M atom, the decisive factor should be the bond strength of M-Al, and the changing trend of the phonon frequency of the B atom is determined by the bond strength of M-B.Among the M 3 AlB 2 phase compounds, M-Al bond exhibits the lowest chemical bond stiffness in each system, with a stiffness value approximately 1/3-1/2 of the corresponding M-B bond stiffness.Additionally, the M-B bond closest to the Al atom demonstrates the highest stiffness.The results in Fig. 5 also indirectly indicate the changing trend of M-Al bond and M-B bond strength.

7 )
peak has both Infrared activity and Raman activity.The schematic diagram of atom vibration, Infrared and Raman spectra of Hf 3 AlB 2 are given in Fig. 8a,b,c, respectively.It can be seen from Fig. 8a that Hf 3 AlB 2 has 6 Infrared modes: E 1u (124.4)I,A 2u (138.3)I,E 1u (153.3)I,A 2u (274.0)I,E 1u (441.7)I,A 2u (531.9)I.Figure 8b shows the position and vibration intensity of the specific peak of the Infrared active vibration mode.E 1u (124.4)I,A 2u (138.3)I,A 2u (274.0)I is clear in the Hf 3 AlB 2 Infrared spectra in Fig. 8b.E 1u (441.7)I,A 2u (531.9)Ihave strong Infrared vibration intensity and larger broadening.However, the Infrared vibration intensity of E 1u (153.3)I is very low and not observable in the study.It can also be seen from Fig. 8a that Ti 3 AlB 2 has 7 Raman active modes: E 2g (70.8)R,E 1g (85.7)R,A 1g (129.5)R,E 2g (151.2)R,E 1g (510.3)R, E 2g (511.7)R,A 1g (591.1)R.Figure 8c gives their specific peak positions and intensities.It can be seen from Fig. 8c that the Raman vibration mode of Hf 3 AlB 2 , Hf 3 AlB 2 has E 2g (70.8)R,E 1g (85.7)R,A 1g (129.5)R,E 2g (151.2) R, E 1g (510.3)R,E 2g (511.7)R, and A 1g (591.1)Rall have high vibration intensity.And the broadening of A 1g (591.1)R peak is larger.While, E 1g (510.3)RE 2g (511.7)Rpeak is very close and almost unrecognizable.The vibration diagram of atoms, Infrared and Raman spectra of Ta 3 AlB 2 are shown in Fig. 9a,b,c, respectively.From Fig. 9a, we can see that Ta 3 AlB 2 also has 6 Infrared modes: E 1u (146.4)I,E 1u (193.3)I,A 2u (202.6)I,A 2u (336.1)I,A 2u (596.9)I,E 1u (659.7)I.Figure 9b shows the position and vibration intensity of the specific peak of the Infrared active vibration mode.E 1u (193.3)I,A 2u (336.1)I,E 1u (659.7)I in Hf 3 AlB 2 structure has strong Infrared vibration intensity.A 2u (202.6)I,A 2u (596.9)Ihave lower Infrared vibration intensity, and the Infrared vibration intensity of E 1u (146.4)I is very low and almost difficult to identify.From Fig.
www.nature.com/scientificreports/accounts for a large proportion at high frequencies.Some of the Infrared or Raman peaks of Zr 3 AlB 2 , Hf 3 AlB 2 , and Ta 3 AlB 2 are difficult to distinguish, probably because the frequencies of the two vibration modes are very close, or because the weak Infrared or Raman peaks are easily annihilated in the substrate spectral lines and are difficult to identify.This paper predicts that the crystal Infrared and Raman spectra of Ti 3 AlB 2 , Zr 3 AlB 2 , Hf 3 AlB 2 , and Ta 3 AlB 2 provide a good theoretical basis for future experiments, but the hyper-Raman spectrum and the change of Raman characteristic spectrum under stress have the potential for further research.